Dimensionless physical constant

"Fundamental physical constant" redirects here. For other uses, see Fundamental physical constant (disambiguation).

In physics, dimensionless or fundamental physical constants are, in the strictest sense, universal physical constants that are independent of systems of units and hence are dimensionless quantities. However, the term may also refer (as in NIST) to any dimensional universal physical constant, such as the speed of light, vacuum permittivity, and the gravitational constant.

Introduction

While both mathematical constants and fundamental physical constants are dimensionless, the latter are determined only by physical measurement and are not defined by any combination of pure mathematical constants.

Physicists try to make their theories simpler and more elegant by reducing the number of physical constants appearing in the mathematical expression of physical theory. They do this by defining the units of measurement in such a way that a number of common physical constants, such as the speed of light, are normalized to unity. The resulting system of units, known as natural units, has a fair following in the literature on advanced physics because it considerably simplifies many equations. When physical quantities are measured in terms of natural units, those quantities become dimensionless.

Certain physical constants, however, are dimensionless numbers which cannot be eliminated in this way. Hence their values must be ascertained experimentally. Perhaps the best known example is the fine structure constant,

$\alpha ={\frac {e^{2}}{\hbar c\ 4\pi \epsilon _{0}}}={\frac {1}{137.03599911}},$

where $e\$ is the elementary charge, $\hbar \$ is the reduced Planck's constant, $c\$ is the speed of light in a vacuum, and $\epsilon _{0}\$ is the permittivity of free space. In simple terms, the fine structure constant determines how strong the electromagnetic force is. There is no accepted theory of why $\alpha$ has the value it does. The analog of the fine structure constant for gravitation is the gravitational coupling constant.

A long-sought goal of theoretical physics is to find first principles from which some or all of the dimensionless constant can be calculated instead of being empirically estimated. The reduction of chemistry to physics was an enormous step in this direction, since properties of atoms and molecules can now be calculated from the Standard Model, at least in principle. The list of fundamental physical constants increases when experiments measure new relationships between physical phenomena. The list decreases when physical theory advances and shows how some previously fundamental constant can be computed in terms of others. A successful Grand Unified Theory or Theory of Everything might yield a further reduction in the number of fundamental constants, ideally to zero. However, this goal remains elusive.

The Standard Model

According to Michio Kaku (1994: 124-27), the Standard Model of particle physics contains nineteen arbitrary dimensionless constants that describe the masses of the particles and the strengths of the various interactions. This was before it was discovered that neutrinos can have nonzero mass, and his list includes a quantity called the theta angle which seems to be zero. After the discovery of neutrino mass, and leaving out the theta angle, John Baez (2002) noted that the new Standard Model requires twenty-five arbitrary fundamental constants, namely the:

• Fine structure constant;
• Strong coupling constant;
• Masses of the fundamental particles (represented in terms of the Planck mass or some other natural unit of mass), namely the six quarks, the six leptons, the Higgs boson, the W boson and the Z boson;
• Four parameters of the CKM matrix, which describe how quarks oscillate between different forms;
• Four parameters of the Maki-Nakagawa-Sakata matrix, which does the same thing for neutrinos.

Gravity requires one more fundamental constant, namely the:

• cosmological constant of Einstein's equations for general relativity.

This makes for a current total of 26 dimensionless fundamental physical constants. More constants presumably await discovery, to describe the properties of dark matter. If the description of dark energy turns out to be more complicated than can be modelled by the cosmological constant, yet more constants will be needed.

Martin Rees's 6 Numbers

In his book Just Six Numbers, Sir Martin Rees mulls over the following numbers, which he considers fundamental to physical theory:

• ν, Nu: the ratio of the coupling constants for the electroweak force and gravitation (circa 10⁴³). The former is simply the fine structure constant, while the latter is the gravitational coupling constant;
• ε, Epsilon: related to the strong force;
• ω, Omega: the number of electrons and protons in the observable universe (circa 10⁸⁰);
• λ, Lambda: the cosmological constant;
• Q: a ratio of fundamental energies;
• δ, Delta: the number of spatial dimensions.

Since Delta must be a nonzero natural number (apparently 3) and is a quantity that we cannot measure, most physicists would not deem it a dimensionless physical constant of the sort discussed in this entry. There are also a number of physical and mathematical reasons, discussed in the entry spacetime, why Delta is 3.

These 6 constants constrain any plausible fundamental physical theory, which must either be able to derive their values from the mathematics of the theory, or accept them as empirical and arbitrary. The question then arises: how many values of these constants result from purely mathematical considerations, and how many represent degrees of freedom for possible valid physical theories, only some of which are possible in a universe, such as ours, with intelligent observers? This leads to a number of interesting possibilities, including the possibility of multiple universes, each with different values of these constants. Multiple universes give rise to selection effects and the anthropic principle.

Other

The mathematician Simon Plouffe has made an extensive search of computer databases of mathematical formulae, seeking formulae giving the mass ratios of the fundamental particles.

The study of the fundamental constants sometimes has bordered on numerology. For instance, the physicist Arthur Eddington argued that for several mathematical reasons, the fine structure constant had to be exactly 1/136. When its value was discovered to be closer to 1/137, he changed his argument to match that value. Experiments since his day have shown that his arguments are still wrong; the constant is about 1/137.036.

See also

• Cabibbo-Kobayashi-Maskawa matrix (Cabibbo angle)
• coupling constant
• Fine-structure constant
• gravitational coupling constant
• Neutrino oscillation
• Physical cosmology
• Standard Model
• Weinberg angle

References

• John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
• John D. Barrow and Frank J. Tipler, 1986. The Anthropic Cosmological Principle. Oxford Univ. Press.
• Michio Kaku, 1994. Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford University Press.
• Martin Rees, 1999. Just Six Numbers: The Deep Forces that Shape the Universe. London: Phoenix. ISBN 0-7538-1022-0

External articles

General

• Fundamental Physical Constants from NIST
• Values of fundamental constants. CODATA, 2002.
• John Baez, 2002, "How Many Fundamental Constants Are There?".
• Plouffe. Simon, 2004, "A search for a mathematical expression for mass ratios using a large database."
• Sheppard, M. R., 2007, "Systematic Search for Expressions of Dimensionless Constants using the NIST database of Physical Constants."

Do the fundamental constants vary?

• John Bahcall, Steinhardt, Charles L., and Schlegel, David, 2004, "Does the fine-structure constant vary with cosmological epoch?" Astrophys. J. 600: 520.
• John Barrow and Webb, J. K., "Inconstant Constants - Do the inner workings of nature change with time?" Scientific American (June 2005).
• Marion, H., et al. 2003, "A search for variations of fundamental constants using atomic fountain clocks," Phys.Rev.Lett. 90: 150801.
• Martins, J.A.P. et al., 2004, "WMAP constraints on varying α and the promise of reionization," Phys.Lett. B585: 29-34.
• Olive, K.A., et al., 2002, "Constraints on the variations of the fundamental couplings," Phys.Rev. D66: 045022.
• Uzan, J-P, 2003, "The fundamental constants and their variation: observational status and theoretical motivations," Rev.Mod.Phys. 75: 403.
• Wandelt, Ben, "Changing Constants, Dark Energy and the Absorption of 21cm Radiation," 2Physics.com, July 25, 2007.
• Webb, J.K. et al., 2001, "Further evidence for cosmological evolution of the fine-structure constant," Phys. Rev. Lett. 87: 091301.

External links

• Murphy, Michael, Web page at the Swinburne University of Technology, Australia.
• Webb, John K., Web page at the University of New South Wales, Australia.